×

zbMATH — the first resource for mathematics

Positivity for cluster algebras from surfaces. (English) Zbl 1331.13017
Summary: We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)] for cluster algebras from surfaces, in geometric type.

MSC:
13F60 Cluster algebras
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05E15 Combinatorial aspects of groups and algebras (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Amiot, C., Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. inst. Fourier, 59, 6, 2525-2590, (2009) · Zbl 1239.16011
[2] Assem, I.; Brüstle, T.; Charbonneau-Jodoin, G.; Plamondon, P.G., Gentle algebras arising from surface triangulations, Algebra number theory, 4, 2, 201-229, (2010) · Zbl 1242.16011
[3] Berenstein, A.; Fomin, S.; Zelevinsky, A., Cluster algebras III: upper bounds and double Bruhat cells, Duke math. J., 126, 1, 1-52, (2005) · Zbl 1135.16013
[4] Buan, A.; Marsh, R.; Reineke, M.; Reiten, I.; Todorov, G., Tilting theory and cluster combinatorics, Adv. math., 204, 572-612, (2006) · Zbl 1127.16011
[5] Caldero, P.; Chapoton, F., Cluster algebras as Hall algebras of quiver representations, Comment. math. helv., 81, 595-616, (2006) · Zbl 1119.16013
[6] Caldero, P.; Chapoton, F.; Schiffler, R., Quivers with relations arising from clusters (\(A_n\) case), Trans. amer. math. soc., 358, 3, 1347-1364, (2006) · Zbl 1137.16020
[7] Caldero, P.; Keller, B., From triangulated categories to cluster algebras, Invent. math., 172, 169-211, (2008) · Zbl 1141.18012
[8] Caldero, P.; Keller, B., From triangulated categories to cluster algebras II, Ann. sci. école norm. sup. (4), 39, 6, 983-1009, (2006) · Zbl 1115.18301
[9] Caldero, P.; Reineke, M., On the quiver Grassmannian in the acyclic case, J. pure appl. algebra, 212, 11, 2369-2380, (2008) · Zbl 1153.14032
[10] Caldero, P.; Zelevinsky, A., Laurent expansions in cluster algebras via quiver representations, Mosc. math. J., 6, 3, 411-429, (2006) · Zbl 1133.16012
[11] G. Carroll, G. Price, Two new combinatorial models for the Ptolemy recurrence, unpublished memo, 2003.
[12] Conway, J.; Lagarias, J., Tiling with polyominoes and combinatorial group theory, J. combin. theory ser. A, 53, 2, 183-208, (1990) · Zbl 0741.05019
[13] Derksen, H.; Weyman, J.; Zelevinsky, A., Quivers with potentials and their representations II: applications to cluster algebras, J. amer. math. soc., 23, 3, 749-790, (2010) · Zbl 1208.16017
[14] Dupont, G., Positivity in coefficient-free rank two cluster algebras, Electron. J. combin., 16, 1, (2009), Research Paper 98, 11 pp · Zbl 1193.16017
[15] Elkies, N.; Kuperberg, G.; Larsen, M.; Propp, J., Alternating-sign matrices and domino tilings I, J. algebraic combin., 1, 2, 111-132, (1992) · Zbl 0779.05009
[16] Felikson, A.; Shapiro, M.; Tumarkin, P., Skew-symmetric cluster algebras of finite mutation type, preprint · Zbl 1262.13038
[17] Fock, V.; Goncharov, A., Moduli spaces of local systems and higher Teichmüller theory, Publ. math. inst. hautes études sci., 103, 1-211, (2006) · Zbl 1099.14025
[18] Fock, V.; Goncharov, A., Dual Teichmüller and lamination spaces, (), 647-684 · Zbl 1162.32009
[19] Fock, V.; Goncharov, A., Cluster ensembles, quantization and the dilogarithm, Ann. sci. école norm. sup. (4), 42, 6, 865-930, (2009) · Zbl 1180.53081
[20] Fomin, S.; Shapiro, M.; Thurston, D., Cluster algebras and triangulated surfaces. part I: cluster complexes, Acta math., 201, 83-146, (2008) · Zbl 1263.13023
[21] S. Fomin, D. Thurston, Cluster algebras and triangulated surfaces. Part II: Lambda lengths, preprint, 2008. · Zbl 07000309
[22] Fomin, S.; Zelevinsky, A., Cluster algebras I: foundations, J. amer. math. soc., 15, 497-529, (2002) · Zbl 1021.16017
[23] Fomin, S.; Zelevinsky, A., Cluster algebras II: finite type classification, Invent. math., 154, 63-121, (2003) · Zbl 1054.17024
[24] Fomin, S.; Zelevinsky, A., Cluster algebras IV: coefficients, Compos. math., 143, 112-164, (2007) · Zbl 1127.16023
[25] S. Fomin, A. Zelevinsky, unpublished result.
[26] Fu, C.; Keller, B., On cluster algebras with coefficients and 2-Calabi-Yau categories, Trans. amer. math. soc., 362, 2, 859-895, (2010) · Zbl 1201.18007
[27] Gekhtman, M.; Shapiro, M.; Vainshtein, A., Cluster algebras and Weil-Petersson forms, Duke math. J., 127, 291-311, (2005) · Zbl 1079.53124
[28] Hernandez, D.; Leclerc, B., Cluster algebras and quantum affine algebras, Duke math. J., 154, 2, 265-341, (2010) · Zbl 1284.17010
[29] Labardini-Fragoso, D., Quivers with potentials associated to triangulated surfaces, Proc. lond. math. soc. (3), 98, 3, 797-839, (2009) · Zbl 1241.16012
[30] Musiker, G., A graph theoretic expansion formula for cluster algebras of classical type, Ann. comb., 15, 1, 147-184, (2011) · Zbl 1233.05118
[31] Musiker, G.; Propp, J., Combinatorial interpretations for rank-two cluster algebras of affine type, Electron. J. combin., 14, 1, (2007), Research Paper 15, 23 pp. (electronic) · Zbl 1140.05053
[32] Musiker, G.; Schiffler, R., Cluster expansion formulas and perfect matchings, J. algebraic combin., 32, 2, 187-209, (2010) · Zbl 1246.13035
[33] Nakajima, H., Quiver varieties and cluster algebras, preprint · Zbl 1223.13013
[34] Palu, Y., Cluster characters for triangulated 2-Calabi-Yau categories, Ann. inst. Fourier, 58, 6, 2221-2248, (2008) · Zbl 1154.16008
[35] Propp, J., Lattice structure for orientations of graphs, (1993), preprint
[36] Propp, J., The combinatorics of frieze patterns and markoff numbers, preprint
[37] Schiffler, R., A cluster expansion formula (\(A_n\) case), Electron. J. combin., 15, (2008), #R64 1 · Zbl 1184.13064
[38] Schiffler, R., On cluster algebras arising from unpunctured surfaces II, Adv. math., 223, 6, 1885-1923, (2010) · Zbl 1238.13029
[39] Schiffler, R.; Thomas, H., On cluster algebras arising from unpunctured surfaces, Int. math. res. not. IMRN, 17, 3160-3189, (2009) · Zbl 1171.30019
[40] Sherman, P.; Zelevinsky, A., Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Mosc. math. J., 4, 4, 947-974, (2004), 982 · Zbl 1103.16018
[41] Thurston, W., Conwayʼs tiling groups, Amer. math. monthly, 97, 757-773, (1990) · Zbl 0714.52007
[42] Zelevinsky, A., Semicanonical basis generators of the cluster algebra of type \(A_1^{(1)}\), Electron. J. combin., 14, 1, (2007), Note 4, 5 pp. (electronic) · Zbl 1144.16015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.